Lo-Res texture blurring in blender
Yes and no. Yes, because this is called "texture filtering" or "texture interpolation." No, because Blender is already doing this.

Your question is basically the opposite of this one, and that answer will be very helpful to you.

Suffice to say, you can change the interpolation mode on the texture node in Cycles, or in the Image Sampling section of Blender Internal. Cubic typically gives the result you want. It's a sharpened version that doesn't look pixelated. "Closest" is pixelated, it doesn't do any blurring at all. "Linear" is a little more blurry than cubic, but that's as blurry as it gets without doing some other image processing, which you can't do within Blender. There's currently no way to blur an image texture before using it in a material.

Lastly, it wouldn't help. If you're blurring the image, you're losing information. If you want to NOT lose that information, you have to have a higher-resolution... which isn't a smaller file. 

The unfortunate reality of computing is that computers are terrible at guessing. If the dataset is incomplete, the computer cannot fill in that data intelligently (unless you're Adobe and have made unholy pacts in order to perform the dark magics known as "content aware fill"). If the incomplete dataset is insufficient, you must add more data which comes at a cost. In the case of Adobe, that cost is unholy liaisons. For the rest of us, that cost is file size.

Hope that helps!